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A177145 E.g.f.: arcsin(x). 10
1, 0, 1, 0, 9, 0, 225, 0, 11025, 0, 893025, 0, 108056025, 0, 18261468225, 0, 4108830350625, 0, 1187451971330625, 0, 428670161650355625, 0, 189043541287806830625, 0, 100004033341249813400625, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

A001818 interspersed with zeros. - Joerg Arndt, Aug 31 2013

a(n) is the number of permutations of n-1 where all cycles have even length.  For example, a(5)=9 and the permutations of 4 elements with only even cycles are (1,2)(3,4); (1,3)(2,4); (1,4)(2,3); (1,2,3,4); (1,2,4,3); (1,3,2,4); (1,3,4,2); (1,4,2,3); (1,4,3,2).

a(n) is the number of permutations on n - 1 elements where there are no cycles of even length and an even number of cycles of odd length. - N. Sato, Aug 29 2013

REFERENCES

L. Comtet and M. Fiolet, Sur les dérivées successives d'une fonction implicite. C. R. Acad. Sci. Paris Ser. A 278 (1974), 249-251.

LINKS

Table of n, a(n) for n=1..26.

Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, p. 12.

FORMULA

G.f.: Q(0)*x^2/(1+x) + x/(1+x), where Q(k) = 1 + (2*k + 1)^2 * x * (1 + x * Q(k+1));  - Sergei N. Gladkovskii, May 10 2013 [Edited by Michael Somos, Oct 07 2013]

E.g.f of a(n + 1), n >= 0, is 1/sqrt(1 - x^2). - N. Sato, Aug 29 2013

If n is odd, a(n) ~ 2*n^(n-1) / exp(n). - Vaclav Kotesovec, Oct 05 2013

E.g.f.: arcsin(x) = x + x^3/(T(0)-x^2), where T(k) = 4*k^2*(1+x^2) + 2*k*(5+2*x^2) +6 + x^2 - 2*x^2*(k+1)*(2*k+3)^3/T(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 13 2013

a(n) = (n-1)! - A087137(n-1). - Anton Zakharov, Oct 18 2016

EXAMPLE

1 is in the sequence because, for k=1, f'(x) = 1/sqrt(1-x^2), and f'(0) = 1.

G.f. = x + x^3 + 9*x^5 + 225*x^7 + 11025*x^9 + 893025*x^11 + ...

MAPLE

n0:= 30: T:=array(1..n0+1): f:=x->arcsin(x):for n from 1 to n0 do:T[n]:=(D(f)(0)):f:=D(f):od: print(T):

MATHEMATICA

a[ n_] := If[ n < 1, 0, If[ EvenQ[n], 0, (n - 2)!!^2]]; (* Michael Somos, Oct 07 2013 *)

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ ArcSin[x], {x, 0, n}]]; (* Michael Somos, Oct 07 2013 *)

PROG

(PARI) Vec( serlaplace( sqrt( 1/(1-x^2) + O(x^55) ) ) )

(PARI) {a(n) = if( n<2, n==1, (n-2)^2 * a(n-2))}; /* Michael Somos, Oct 07 2013 */

(PARI) a(n) = if( n<0, 0, n! * polcoeff( asin(x + x * O(x^n)), n)); /* Michael Somos, Oct 07 2013 */

CROSSREFS

Alternate terms are A001818. - N. Sato, May 13 2010

Cf. A087137.

Sequence in context: A339488 A157309 A215484 * A178912 A191564 A067479

Adjacent sequences:  A177142 A177143 A177144 * A177146 A177147 A177148

KEYWORD

nonn

AUTHOR

Michel Lagneau, May 03 2010

STATUS

approved

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Last modified August 1 17:38 EDT 2021. Contains 346402 sequences. (Running on oeis4.)